Stationary distribution and near-optimal control of a stochastic reaction–diffusion cholera model
摘要
Cholera is an acute diarrheal disease caused by the ingestion of food or water contaminated with the bacterium Vibrio cholerae. In this paper, we develop a stochastic reaction–diffusion cholera model that incorporates stochastic noise and spatial diffusion to investigate the dynamic behavior and optimal control of the disease. Firstly, we prove the existence and uniqueness of stationary distribution for the model according to an appropriate Lyapunov functional. Secondly, we introduce three control variables, namely, vaccination control, the treatment of the infected individuals, and vibrio mortality rate via water sanitation, into the stochastic reaction–diffusion cholera model to derive an optimal control model. In both theoretical framework and practical applications, near-optimal controls are as significant as optimal controls. Thus, we establish the sufficient and necessary conditions for the near-optimal control system of the stochastic cholera model in terms of Pontryagin’s stochastic maximum principle, adjoint equations, and some prior estimates. Finally, numerical simulations are conducted to validate that the three controls can effectively reduce the number of infected individuals, meanwhile, moderately increasing the intensity of stochastic noise and restricting the mobility of susceptible and infected individuals are effective strategies for preventing and controlling cholera outbreaks.